Derivative Sum for One-Slot Energy Terms

Proved

FunctionalInequalities.Analysis.CoordinateTensor.slot_derivative_sum_readable

by Lazyfill · Oct 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

functional-inequalitieskls-conjecturekls-modularkls-readablelog-concave

This auxiliary lemma evaluates the first-order matrix contribution in a coordinate-tensor energy calculation.

For an order- $r$ coordinate tensor $T$ and a real matrix $M$ , the finite sum

\sum_{J,K}T_JT_K\sum_s \left(\prod_{u\ne s}\delta_{J_u,K_u}\right)(-M_{J_sK_s})

equals

-\langle T,\Sigma_M T\rangle.

The result is a purely finite algebraic identity used internally in differentiating the energy form.

Original Lean declaration header

Within namespace FunctionalInequalities.Analysis.CoordinateTensor. Shown in its original module context. Proof omitted.

private lemma slot_derivative_sum
    (T : (Fin r → Fin n) → ℝ) (M : Matrix (Fin n) (Fin n) ℝ) :
    (∑ J : Fin r → Fin n, ∑ K : Fin r → Fin n,
      T J * T K * (∑ s : Fin r,
        (∏ u ∈ Finset.univ.erase s, if J u = K u then (1 : ℝ) else 0) *
          (-M (J s) (K s)))) =
      -inner T (actSum M T)

Module navigation

Module: Analysis/Tensor/EnergyForm.

Definition modules: Analysis/Tensor/SlotAction.

Supporting theorems: sum_mul_prod_delta_erase.

Previous: sum_mul_prod_delta_erase · Next: hasDerivAt_energyForm_line.

Source excerpt, lines 233–239 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_FunctionalInequalities_Analysis_Tensor_EnergyForm

variable {n r : ℕ}

Formal statement

 lemma FunctionalInequalities.Analysis.CoordinateTensor.slot_derivative_sum_readable
    (T : (Fin r → Fin n) → ℝ) (M : Matrix (Fin n) (Fin n) ℝ) :
    (∑ J : Fin r → Fin n, ∑ K : Fin r → Fin n,
      T J * T K * (∑ s : Fin r,
        (∏ u ∈ Finset.univ.erase s, if J u = K u then (1 : ℝ) else 0) *
          (-M (J s) (K s)))) =
      -inner T (actSum M T) := by sorry

Source

Lean source: https://github.com/human-ai-mathematics/kls-lean/blob/408915f/formalization/FunctionalInequalities/Analysis/Tensor/EnergyForm.lean#L233-L291 (repository human-ai-mathematics/kls-lean, commit 408915f, module `FunctionalInequalities.Analysis.Tensor.EnergyForm`, declaration `_private.FunctionalInequalities.Analysis.Tensor.EnergyForm.0.FunctionalInequalities.Analysis.CoordinateTensor.slot_derivative_sum`; the repository is private at upload time).

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